What are the differences between the two gamma settings
The main difference is that the gamma on the color management section will be applied globally at the end of all processes, and to the whole rendered image. It will affect all output images (except for those exported as OpenEXR), including those coming from the compositor, the VSE and even images that do not use the compositor.

Images saved from the UV/Image editor will also be affected by this setting if they are saved using the "Save as render" option.

The gamma node in the compositor will only affect those inputs connected directly to it.